Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Abstract

Free vibration of statically thermal postbuckled functionally graded material (FGM) beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electromechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

“…It is well known that the EBBT, or the classical beam theory, is the simplest one, in which it is assumed that the cross-section is perpendicular to the bending line so that the transverse shear deformation is neglected [1][2][3][4][5][6][7][8][9] . However, in the TBT [10][11][12][13][14][15][16][17][18][19] or the first-order shear deformation beam theory, it allows a rotation between the cross-section and the bending line and takes into account rotational inertia effects.…”

“…Equations (25)-(27) consist of a standard system of three second-order ordinary differential equations including the frequency parameter Ω in terms of the unknown shape functions of U (ξ), W (ξ), and Φ(ξ), which can be numerically solved by the shooting method [7,19] under specified boundary conditions. In order to write dimensionless natural boundary conditions, dimensionless resultant forces and bending moment corresponding to the mode shapes are given,…”

“…On the basis of the EBBT, some authors studied vibration the responses of FGM beams by different analytical and numerical approaches [1][2][3][4][5][6][7][8][9] . Alshorbagy et al [1] investigated free vibration of an FGM beam by the finite element method.…”

“…Li and Liu [6] showed that the deflections, the critical buckling loads, and the natural frequencies of FGM beams are proportional to those of the reference homogenous beams and proved that these relationships are independent of the load and boundary conditions. Li et al [7] performed a small free vibration of thermo-electrically post-buckled FGM beams with surface-bonded piezoelectric layers in thermal environment by a numerical shooting method based on the exact geometrically nonlinear theory for the axially extensible beams. Huang and Li [8] studied free vibration of axially functionally graded beams with non-uniform cross-section using an integration technique to transform the differential governing equations into Fredholm integral equations.…”

Chaotic vibrations of flexible curvilinear beams in temperature and electric fields

“…The post-buckling behavior of functionally graded material beams with edge crack effects was investigated by Ke et al [11]. Li et al [12] analyzed thermal stability of nonlinear vibrations of a hybrid functionally graded Timoshenko beam with both clamped edges.…”